The variation method and the algebraic eigenvalue problem

The generalized algebraic eigenvalue problem ( A ? λ B )x = 0 arises in the use of the variation method in quantum mechanics. If, within the limitations of the computer word-length, the basis set used to expand the trial wave function is linearly dependent, the matrix B becomes singular. Three different algorithms designed to deal with this difficulty have been investigated, paying special attention to the problem of identifying which members of the basis set are effectively linearly dependent. The advantages and limitations of each method are discussed.     

The eigenvalue problem in phase space

We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c‐function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi‐representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc.     

Excited-state energy eigenvalue and wave-function evaluation of the Gaussian asymmetric double-well potential problem via numerical shooting method 2

This project aims at computation excited-state energy eigenvalues and wave-function of a particle under Gaussian asymmetric double-well potential using numerical shooting method and perturbation theory a method to deal with discrete-eigenvalue problems. We also compare the energy eigenvalue and wave-function with those obtained from other typical means popular among physics students, namely the numerical shooting method and perturbation theory. Show that the idea of program of the numerical shooting method and perturbation theory of this problem (see Sects. 2.1 and 4) The numerical shooting method is generally regarded as one of the most efficient methods that give very accurate results because it integrates the Schr?dinger equation directly, though in the numerical sense. The n = even case is shown in Figs. 4, 5 and 6. In this case, the wave-function has split up on asymmetric nodes under Gaussian asymmetric double-well potential. The n = odd case is shown in Fig. 7. In this case, the wave-function has not split up on asymmetric nodes under Gaussian asymmetric double-well potential.     

Excited-state energy eigenvalue and wave-function evaluation of the Gaussian symmetric double-well potential problem via numerical shooting method 1

This work aims at computing excited-state energy eigenvalues and wave-function of a particle under Gaussian symmetric double-wells potential using numerical shooting method and perturbation theory a method to deal with discrete-eigenvalue problems. We also compare the energy eigenvalue and wave-function with those obtained from other typical means popular among physics students, namely the numerical shooting method and perturbation theory. Show that the idea of program of the numerical shooting method and perturbation theory of this problem (see Sects. 2.2 and 3). The numerical shooting method is generally regarded as one of the most efficient methods that give very accurate results because it integrates the Schrödinger equation directly, though in the numerical sense. The n = even case is shown in Fig. 5. In this case, the wave-function has split up on symmetric nodes under Gaussian symmetric double-well potential.     

An iterative method to solve the algebraic eigenvalue problem

An iterative method based on perturbation theory in matrix form is described as a procedure to obtain the eigenvalues and eigenvectors of square matrices. Practical vector notation and elementary schematic algorithm codes are given. The particular programming characteristics of the present computational scheme are based upon eigenvector corrections, obtained through a simple Rayleigh–Schrödinger perturbation theory algorithm. The proposed methodological processes can be used to evaluate the eigensystem of large matrices.     

Chiral diphosphites derived from d-glucose in the copper-catalyzed conjugate addition of diethylzinc to cyclohexenone

A series of diphosphite ligands 1–3 derived from readily available d-(+)-glucose and bisphenol or binaphthol derivatives have been applied as ligands in the Cu-catalyzed 1,4-addition of diethylzinc to cyclohexenone. Excellent reaction rates (TOF>1200 h⁻¹) and good enantioselectivities (e.e. of up to 84%) were achieved. The modular nature of these ligands allows easy systematic variation in the configuration of the stereocenters (C(3), C(5)) at the ligand backbone and in the biaryl substituents, so the optimum configuration for maximum enantioselectivity in the asymmetric 1,4-addition can be determined. The results obtained show that the enantioselectivity induced by the ligand depends strongly on the absolute configuration of the C(3) stereogenic centre, while the sense of the enantiodiscrimination is predominantly controlled by the configuration of the biaryl groups of the phosphite moieties.     

Fragmentation of conjugate bases of esters derived from multifunctional alcohols including triacylglycerols

Enolate anions of esters from 1,2 and 1,3 diols undergo an internal nucleophilic substitution reaction that produces a β-ketoester and an alkoxide ion within the molecular species. These intermediate ions undergo two competitive fragmentation pathways. The first pathway corresponds to a second nucleophilic substitution of the ketoester by the alkoxide that yields a neutral cyclic ether and the β-ketoacid carboxylate. The latter then loses carbon dioxide and produces the enolate anion of the corresponding ketone. The second proposed pathway is stepwise: it starts with a proton transfer from the methylene group between the two carbonyls to the alkoxide anion that produces an alcohol and the enolate ion of the β-ketoester inside the molecular species. The latter undergoes cleavage of the ester bond induced by the negative charge to yield an ion-dipole complex composed of a neutral acylketene and an alkoxide ion. The direct dissociation of this ion-dipole complex competes with an internal proton exchange to yield a new complex that consists of an alcohol molecule and the anion of the acylketene, which can also dissociate. The fragmentation pathway that leads to the ketone enolate is sensitive to the relative positions (1,2 or 1,3) of the esters on the molecular backbone. This position-sensitive reaction is useful for the assignment of the primary and secondary positions in triacylglycerols, even in mixtures, as shown by some examples.     

Some upper bounds for the energy of graphs

Let G = (V,E) be a graph with n vertices and e edges. Denote V(G) = {v ₁,v ₂,...,v _n }. The 2-degree of v _i , denoted by t _i , is the sum of degrees of the vertices adjacent to . Let σ _i be the sum of the 2-degree of vertices adjacent to v _i . In this paper, we present two sharp upper bounds for the energy of G in terms of n, e, t _i , and σ _i , from which we can get some known results. Also we give a sharp bound for the energy of a forest, from which we can improve some known results for trees.     

Self-aggregation of reverse bis peptide conjugate derived from the unstructured region of the prion protein

A novel bis scaffold containing a pentapeptide PHGGG, derived from the prion octarepeat, forms fibrils of nanometric dimensions, thus indicating that certain segments in the unstructured region of the prion protein may facilitate an initial fibrillation event.     

Stereoselective conjugate addition of benzyl phenylsulfonyl carbanions to enoates derived from d-mannitol

The conjugate addition of benzylic phenylsulfonyl carbanions (2a'-d') to enoates derived from d-(+)-mannitol (E- or Z-1a-c) was studied using THF and THF/HMPA as solvent. Under kinetic conditions (-78 degrees C), enoate E-1a,b led to a mixture of syn-(R,S) and anti-(S,S) adducts (55/45), and syn-(R,S) adducts were the main product obtained ( approximately 90/10) from enoate Z-1a. Under thermodynamic conditions (-78 degrees C to room temperature) syn-(R,S) adducts were also preferentially formed ( approximately 90/10), despite the geometry at the double bond in the acceptor. Enoate 1c (E/Z = 57/43), bearing an additional benzyl group at the alpha-position, also reacted with carbanions 2'a,b, under thermodynamic conditions, leading to syn-adducts in excellent de (control at the three newly generated stereogenic centers). The adducts were quantitatively transformed into the corresponding beta-gamma-disubstituted gamma-butyrolactones and alpha,beta,gamma-trisubstituted gamma-butyrolactones. (1)H NMR studies (NOE and J-coupling) of these lactones allowed us to determine their configuration at the newly generated chiral centers. The reduction of the C-S bond in adducts syn-(R,S) with Na/Hg, followed by treatment of the resulting products in aqueous acid media, led to enantioenriched beta-benzyl-gamma-hydroxymethyl-gamma-butyrolactones. The conformational equilibrium of enoates E- and Z-1b was evaluated by theoretical calculations (ab initio, MP2/6-31G), and a mechanistic rationale was proposed to explain the observed stereoselectivities.     

Rigorous energy bounds for two-electron systems

Rigorous lower and upper bounds on the lowest three eigenvalues of the two-electron atomic Hamiltonian are obtained for the symmetry sectors ¹S,³S,¹P,³P and the sector ³P^o with unnatural parity. The bounds result from three-dimensional projection operators and are given as explicit expressions that depend on the nuclear charge Z as parameter. They are designed for application within further analysis, and we exemplify this by demonstrating monotonicity properties of excitation energies.     

Constrained solutions of the eigenvalue problem in truncated basis sets

It is shown that simple orthogonality constraints between some set of known approximate eigenfunctions and another set of functions which are to be determined as approximate eigensolutions need to be modified. The proposed modification introduces a measure of the approximate character of the known functions and leads to the reduction of the dimensionality of the eigenvalue problem for other solutions. The discussed method is fully variational and leads directly to a Hermitian eigenvalue problem. This approach is also independent of the choice of truncated basis sets for different classes of approximate solutions of the eigenvalue problem. © 1997 John Wiley & Sons, Inc.     

An analysis of the siegert eigenvalue problem for autoionizing states

We show that the divergent integrals which appear in a direct matrix solution to the Siegert problem for autoionizing (or electron scattering) state energies and widths can be cancelled exactly. When this is done the Siegert problem becomes essentially a bound state problem. We also show that the resulting non-hermitian secular equation which requires several non-hermitian diagonalizations in the iterative solution for the complex energy can be exactly reduced by a partitioning technique to a single hermitian diagonalization (for a single open channel) with the subsequent iterative solution of a simple algebraic equation.     

Geminals in Dirac–Coulomb Hamiltonian eigenvalue problem

The applicability of the Dirac–Coulomb model in computational analysis of the properties of many-electron systems has been, since many years, a subject of dispute and controversy. The most common and numerically safe approach, based on the restriction of the variational space to the many-electron spinors spanning a subspace of the positive-energy part of the complete Hilbert space has been challenged by alternative models in which carefully selected both positive and negative energy functions are taken into account. However, these constructions are not possible when one goes beyond the one-electron model, e.g. when geminal-containing trial functions are used. Then the problem becomes particularly difficult and subtle. In this report several aspects specific for the geminal-based variational approach to the Dirac–Coulomb eigenvalue problem are discussed.     

On the Ricatti equation for eigenvalue problems

The Riccati equation is shown to be suitable for obtaining implicit approximate analytic expressions for the eigenvalues of quantum-mechanical systems. The Hamiltonian operator H = (1/2)p² – (Z/r) + λr² is used as a test example, and the resulting formulae are modified to deal with the Zeeman effect in hydrogen.     

Lower bounds for the energy levels of the lithium atom

A recent modification of the intermediate hamiltonian method, intended to circumvent the problem associated with a low-lying continuum in the operator H⁰, has been applied to the lithium atom with a basis of discrete hydrogenic functions. The intended raising of the continuum occurs, but not to the extent that the beginning of the continuum is raised above the ground state of the atom; indications are that no basis of discrete hydrogenic eigenfunctions can accomplish this. Since these calculations were performed on the full lithium atom hamiltonian, they are not strictly comparable to those of Fox and Siglillito, in which a simplified hamiltonian was used.     

The Sherman equations as a nonlinear Perron eigenvalue problem

When a chemical sample composed of N elements is analyzed using sequential selective excitation by a tunable polyenergetic X-ray beam and selective measurement of the characteristic X-rays, the production of secondary fluorescence does not interfere with the measurements. This experimental situation leads to a particular case of the Sherman equations which can be written as a set of non-linear equations. The same kind of equations are also obtained when we excite a chemical sample with a polyenergetic X-ray beam and neglect the production of secondary fluorescence. This set of equations can be regarded as a non-linear eigenvalue problem. A non-linear extension of the Perron Frobenious theorem ensures that there is one and only one physically acceptable solution, and also leads to a method to obtaining it. The propagation off measurements errors of sample fluorescence to errors in the calculated sample concentrations, has been simulated, and the results show that the solution is well conditioned. The case of production of secondary fluorescence can not be treated, in general, as a nonlinear Perron eigenvalue problem, but it has been shown that it is rather plausible that Sherman equations corresponding to the actual chemical elements and that include the production of secondary fluorescence have one and only one physically acceptable solution. An exhaustive search could elucitate the existence and unicity of solutions for the equations corresponding to the actual chemical elements.